Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking and communication. In this paper, we first show that if positive integer k cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order 4n and weight k, where n is an odd positive integer. Then we show that for any square k, there is an integer N(k) such that for each n≥ N(k), there is a symmetric weighing matrix of order n and weight k. Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita and Seberry.
Let G be a group and S a nonempty subset of G. Then, S is product-free if ab∉S for all a,b∈S. We say S is a locally maximal product-free set if S is product-free and not properly contained in any other product-free set. It is natural to ask whether it is possible to determine the smallest possible size of a locally maximal product-free set in G. Alternatively, given a positive integer k, one can ask the following: what is the largest integer nk such that there is a group of order nk with a locally maximal product-free set of size k? The groups containing locally maximal product-free sets of sizes 1 and 2 are known, and it has been conjectured that n3=24. The purpose of this paper is to prove this conjecture and hence show that the list of known locally maximal product-free sets of size 3 is complete. We also report some experimental observations about the sequence nk.
A generalized binomial theorem is developed in terms of Bell polynomials and by applying this identity some sums involving inverse binomial coefficient are calculated. A technique is derived for calculating a class of hypergeometric transformation formulas and also some curious q series identities.
We propose a procedure of constructing new block designs starting from a given one by looking at the intersections of its blocks with various sets and grouping those sets according to the structure of the intersections. We introduce a symmetric relationship of friendship between block designs built on a set $V$ and consider families of block designs where all designs are friends of each other, the so-called friendly families. We show that a friendly family admits a partial ordering. Furthermore, we exhibit a map from the power set of $V$, partially ordered by inclusion, to a friendly family of a particular type which preserves the partial order.
A graph G is said to be a self-centered graph if the eccentricity of every vertex of the graph is the same. In other words, a graph is a self-centered graph if radius and diameter of the graph are equal. In this paper, self-centeredness of strong product, co-normal product, and lexicographic product of graphs is studied in detail. The necessary and sufficient conditions for these products of graphs to be a self-centered graph are also discussed. The distance between any two vertices in the co-normal product of a finite number of graphs is also computed analytically.
We define a two-player combinatorial game in which players take alternate turns; each turn consists of deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player’s move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph’s automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on random graphs. When analyzing Grim played on paths, using the Sprague-Grundy function, we find a connection to a standing open question about Octal games.
We show that the Euler-Mascheroni constant γ and Euler’s number e can both be represented as a product of a Riordan matrix and certain row and column vectors.
This paper considers the varietal hypercube network VQn with mixed faults and shows that VQn contains a fault-free Hamilton cycle provided faults do not exceed n-2 for n⩾2 and contains a fault-free Hamilton path between any pair of vertices provided faults do not exceed n-3 for n⩾3. The proof is based on an inductive construction.
A graph G is said to be even if all vertices of G have even degree. Given a k-edge-coloring of a graph G, for each color i∈Zk={0,1,…,k-1} let G(i) denote the spanning subgraph of G in which the edge-set contains precisely the edges colored i. A k-edge-coloring of G is said to be an even k-edge-coloring if for each color i∈Zk, G(i) is an even graph. A k-edge-coloring of G is said to be evenly-equitable if for each color i∈Zk, G(i) is an even graph, and for each vertex v∈V(G) and for any pair of colors i,j∈Zk, |degG(i)(v)-degG(j)(v)|∈{0,2}. For any pair of vertices {v,w} let mG({v,w}) be the number of edges between v and w in G (we allow v=w, where {v,v} denotes a loop incident with v). A k-edge-coloring of G is said to be balanced if for all pairs of colors i and j and all pairs of vertices v and w (possibly v=w), |mG(i)({v,w})-mG(j)({v,w})|≤1. Hilton proved that each even graph has an evenly-equitable k-edge-coloring for each k∈N. In this paper we extend this result by finding a characterization for graphs that have an evenly-equitable, balanced k-edge-coloring for each k∈N. Correspondingly we find a characterization for even graphs to have an evenly-equitable, balanced 2-edge-coloring. Then we give an instance of how evenly-equitable, balanced edge-colorings can be used to determine if a certain fairness property of factorizations of some regular graphs is satisfied. Finally we indicate how different fairness notions on edge-colorings interact with each other.
We derive a formula for the reliability of a d-dimensional consecutive-k-out-of-n:F system. That is, a formula for the probability that an n_1 ×…× n_d array whose entries are (independently of each other) 0 with probability p and 1 with probability q = 1 - p does not include a contiguous s_1 ×…× s_d subarray whose every entry is 1.
Much research has involved the consideration of graphs which have subgraphs of a particular kind, such as cliques. Known classes of graphs which are eigen-bi-balanced, that is, they have a pair a, b of nonzero distinct eigenvalues, whose sum and product are integral, have been investigated. In this paper we will define a new class of graphs, called q-cliqued graphs, on q2+1 vertices, which contain q cliques each of order q connected to a central vertex, and then prove that these q-cliqued graphs are eigen-bi-balanced with respect to a conjugate pair whose sum is -1 and product 1-q. These graphs can be regarded as design graphs, and we use a specific example in an entomological experiment.
A graph on 2n vertices can be starter-labelled, if the vertices can be given labels from the nonzero elements of the additive group Z2n+1 such that each label i, either i or i-1, is assigned to exactly two vertices and the two vertices are separated by either i edges or i-1 edges, respectively. Mendelsohn and Shalaby have introduced Skolem-labelled graphs and determined the conditions of k-windmills to be Skolem-labelled. In this paper, we introduce starter-labelled graphs and obtain necessary and sufficient conditions for starter and minimum hooked starter labelling of all k-windmills.
The n-dimensional hypercube Qn is bipancyclic; that is, it contains a cycle of every even length from 4 to 2n. In this paper, we prove that Qn (n≥3) contains a 3-regular, 3-connected, bipancyclic subgraph with l vertices for every even l from 8 to 2n except 10.
Let R be a commutative finite principal ideal ring with unity, and let G(R) be the simple graph consisting of nontrivial proper ideals of R as vertices such that two vertices I and J are adjacent if they have nonzero intersection. In this paper we continue the work done by Abu Osba. We calculate the radius, eccentricity, domination number, independence number, geodetic number, and the hull number for this graph. We also determine when G(R) is chordal. Finally, we study some properties of the complement graph of G(R).
The association of integers, conjugate pairs, and robustness with the eigenvalues of graphs provides the motivation for the following definitions. A class of graphs, with the property that, for each graph (member) of the class, there exists a pair of nonzero, distinct eigenvalues, whose sum and product are integral, is said to be eigen-bibalanced. If the ratio is a function , of the order of the graphs in this class, then we investigate its asymptotic properties. Attaching the average degree to the Riemann integral of this ratio allowed for the evaluation of eigen-balanced areas of classes of graphs. Complete graphs on vertices are eigen-bibalanced with the eigen-balanced ratio which is asymptotic to the constant value of −1. Its eigen-balanced area is —we show that this is the maximum area for most known classes of eigen-bibalanced graphs. We also investigate the class of eigen-bibalanced graphs, whose class of complements gives rise to an eigen-balanced asymptote that is an involution and the effect of the asymptotic ratio on the energy of the graph theoretical representation of molecules.
Let R be a commutative ring with identity. The zero-divisor graph of R, denoted Γ(R), is the simple graph whose vertices are the nonzero zero-divisors of R, and two distinct vertices x and y are linked by an edge if and only if xy=0. The genus of a simple graph G is the smallest integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine that the genus of the zero-divisor graph of Zn, the ring of integers modulo n, is two or three.
Every ordinary Riordan array contains two naturally embedded Riordan arrays. We explore this phenomenon, and we compare it to the situation for certain moment matrices of families of orthogonal polynomials.
Let n≥2 be a positive integer and q a prime power. Consider necklaces consisting of n beads, each of which has one of the given q colors. A primitive Cn-orbit is an equivalence class of n necklaces closed under rotation. A Cn-orbit is self-complementary when it is closed under an assigned color matching. In the work of Miller (1978), it is shown that there is a 1-1 correspondence between the set of primitive, self-complementary Cn-orbits and that of self-reciprocal irreducible monic (srim) polynomials of degree n. Let N be a positive integer relatively prime to q. A q-cycle mod N is a finite sequence of nonnegative integers closed under multiplication by q. In the work of Wan (2003), it is shown that q-cycles mod N are closely related to monic irreducible divisors of xN-1∈Fq[x]. Here, we show that: (1) q-cycles can be used to obtain information about srim polynomials; (2) there are correspondences among certain q-cycles and Cn-orbits; (3) there are alternative proofs of Miller's results in the work of Miller (1978) based on the use of q-cycles.
Let T0(n,k) be the number of all labeled T0-topologies having k open sets that we can define on n points, and let t0(n,k) be the number of those which are nonhomeomorphic. In this paper, we compute these numbers for k≥5·2n-4 and arbitrary n≥4. The numbers tn0(n,k) of all unlabeled and non-T0-topologies with k open sets are also given for k≥2n-2.
